probabilistic network theory
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Probabilistic network theory
Philippa Pattison & Garry Robins1
Department of Psychology
University of Melbourne
Introduction
The aim of this chapter is to describe the foundations of probabilistic network theory. We review
the development of the field from an early reliance on simple random graph models to the
construction of progressively more realistic models for human social networks. We hence show
how developments in probabilistic network models are increasingly able to inform our
understanding of the emergence and structure of social networks in a wide variety of settings.
Social networks
Growing numbers of social scientists from an increasingly diverse set of disciplines are turning their
attention to the study of social networks. The precise reasons vary, but almost certainly there are at
least two key factors at work. The first is an increasing recognition that networks matter in many
realms of social, political and economic life. Networks both potentiate and constrain the social
interactions that, for instance, underpin the dissemination of knowledge, the exercise of power and
influence, and the transmission of communicable diseases. Ignoring the structured nature of these
interactions often leads to erroneous conclusions about their consequences, as social scientists from
a number of disciplines have repeatedly pointed out (e.g., Bearman, Moody & Stovel, 2004;
Kretzschmar & Morris, 1996). The second reason for a heightened focus on social networks is our
increasing capacity to measure, monitor and model social networks and their evolution through
time, and hence to draw social networks into a more general program for a quantitative social
science. Probabilistic models for social networks have played and will likely increasingly play
a vital role in these developments. In this chapter we review progress in attempts to develop
probabilistic network models, and point to areas of ongoing development.
What is distinctive about models for social networks?
At the outset it is important to recognise that social networks pose particular challenges as far as
probability modelling is concerned. Unlike observations on a set of distinct actors, where an
1We are grateful to Peng Wang and Galina Daraganova for helpful comments on this chapter.
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assumption of independent observations may often seem reasonable, social relationships are much
less plausibly regarded as independent. Relational observations may share one or more actors and
hence be subject to such influences as the goals and constraints of a particular actor. Alternatively,
they may be linked by other relationships (eg the relationship between actors i andj may be linked
with the relationship between actors kand l by a relationship involving actorsj and k) and hence
dependent, for example, by virtue of competition or cooperation regarding relational resources
involving actorsj and k. As we see below, the development of probabilistic network models began
with simple models that assumed independent relational ties, but empirical researchers quickly
confronted the problem that social networks appeared to deviate from simple random structures in
seemingly systematic ways. Hence, the story of the development of probabilistic network models is
a story of alternatively probing and parameterising progressively more complex systematicities in
network structure.
Notation and some basic properties of graphs and directed graphs
We begin with some notation and some important definitions, referring the reader to Wasserman
and Faust (1994; also Bollobs, 1998) for a fuller exposition of key concepts.
Graphs and directed graphs
We letN= {1,2,,n} be a set of networknodes, with each node representing a social actor. The
actors are often persons, but may also be groups, organisations or other social entities. An observed
social network may be represented as a graphG = (N,E) comprising the node setNand the edge set
Ecomprising all pairs (i,j) of distinct actors who are linked by a network tie. The tie, or edge, (i,j)
is said to be incidentwith nodes i andj. In this case, the network ties are taken to be nondirected,
with no distinction between the tie from actori to actorj and the tie from actorj to actori; in other
words, the edges (i,j) and (j,i) are regarded as indistinguishable. If it is desirable to distinguish
these ties as it often is the network may be represented instead by a directed graph (N,E) onN:
the node set is then alsoN, and the arc setEis the set of all orderedpairs (i,j) such that there is a tie
from actorito actorj. The convention of using the term edge in the case of a non-ordered pair (that
is, a nondirectional tie) and arc in the case of an ordered pair (that is, a directional tie) is widely
adopted by graph theorists, although network researchers are inclined to use the term tie
interchangeably in both cases. In many cases, by convention, ties of the form (i,i), known as loops,
are excluded from consideration.
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Graphs and directed graphs can be conveniently represented by a graph drawing. The elements of
the node setNare represented by points in the drawing, and a nondirected line connects node i and
nodej if (i,j) is an edge in the edge setE. In the case of a directed graph, an arc represented by a
directed arrow is drawn from node i to nodej if (i,j) is in the arc setE. Figures 1 and 2 show
examples of a graph and directed graph, respectively.
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Insert Figures 1 and 2 about here
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Order, size and density
The orderand size of a graph are defined to be the number of nodes and edges, respectively;
likewise the order and size of a directed graph are the number of nodes and arcs, respectively. Forexample, the graph in Figure 1 has order 17 and size 34; the directed graph in Figure 2 has order 37
and size 169. The size of a graph of ordern varies between 0 for an empty graph and n(n-1)/2 for
the complete graph of ordern (that is, the graph in which every pair of nodes is linked by an edge).
For a directed graph of ordern, size varies between 0 and n(n-1). The density of a graph or directed
graph is a ratio of its actual size to the maximum possible size forn nodes, and is in the range [0,1].
The densities of the graph and directed graph of Figures 1 and 2 are 0.25 and 0.13, respectively.
Adjacency matrix
Graphs and directed graphs can be represented by a binary adjacency matrix. For example, in the
case of a graph, we can define x to be an n n matrix with entries:xij = 1 if there is an edge
between i andj; andxij = 0, otherwise. Sincexij = 1 if and only ifxji = 1, x is necessarily a
symmetric matrix. In the case of a directed graph, unit entries in x correspond to arcs inE(that is,
xij = 1 if and only if there is an arc from i andj; andxij = 0, otherwise). In this case, symmetry is not
necessarily implied. The adjacency matrix corresponding to the graph of Figure 1 is shown in
Table 1. Note that the size of a graph isx++/2 whereas the size of a directed graph isx++, wherex++
= i,jxij is the sum of entries in the adjacency matrix.
Degree, degree sequence and degree distribution
The numberk(i) of edges incident with a given node i is termed its degree: k(i) = jxij is the sum of
row i in the adjacency matrix x. The degree sequence (k(1), k(2), , k(n)) of a graph is the
sequence of the degrees of its nodes, indexed by the labels 1,2,,n of nodes inN. The degree
distribution is (d0, d1,, dn-1), where dkis the number of nodes in G of degree k. For example, the
degree distribution of the graph in Figure 1 is shown in Figure 3. In a directed graph, the concept of
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degree is more complex, since there may be an arc directed from nodej towards a given node i or
away from node i towards nodej, or there may be arcs in both directions between nodes i andj. We
therefore characterise each node i in a directed graph by its: outdegree outi =xi+ = jxij; indegree ini
=x+i = jxji; and mutual degree muti = jxijxji.
-----------------------------------------------
Insert Figure 3 and Table 1 about here
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Subgraphs and the dyad and triad census
Each subset S of the node setNof a graph G = (N,E) gives rise to an induced subgraphHofG with
node set S and edge setE containing all edges in G that link pairs of nodes in S. More generally,
any graphH=(N,E) is a subgraph ofG ifNNandEE. For example, the subgraph induced
by the node set {1,6,13} of the graph of Figure 1 has edge set {(1,6), (1,13), (6,13)}; the graph
comprising the node set {1,6,13} and the edge set {(1,6), (1,13)} is a subgraph of the graph of
Figure 1, but not an induced subgraph. If every pair of nodes in the subgraphHis connected by an
edge, thenHis said to be a clique; for example, the subgraph induced by {2,7,8,9,10,11} in Figure
1 is a clique of order 6.
A useful set of descriptive statistics for a graph or directed graph is a summary of the form of all ofits small subgraphs. For example, the dyad census is a count of the number of each possible type of
2-node induced subgraphs and the triad census is the set of counts of 3-node induced subgraphs.
For example, the graph of Figure 1 has 102 null dyads and 34 linked dyads; its triad census
comprises 279, 322, 49 and 30 induced 3-node subgraphs with 0, 1, 2 and 3 edges, respectively.
The dyad census and the triad census for directed graphs are defined similarly, but the number of
forms of 2- and 3-node subgraphs is greater in the directed graph case.
Implicit in the description of the dyad and triad census is the notion that two graphs (or subgraphs)
can have the same form. We can make this notion more explicit by defining an isomorphic
mapping between graphs. Specifically, two graphs G =(N,E) andH=(N,E) are isomorphic if there
is a one-to-one mapping fromNontoN such that (i,j) is an edge inEif and only if ((i),(j)) is
an edge inE.
Paths, reachability and connectedness
Social networks are often important to understand because social processes such as the diffusion
of information, the exercise of influence, and the spread of disease are potentiated by network
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this set contains 2n(n-1)/2 graphs, since each of the n(n-1)/2 pairs of nodes may or may not be linked
by an edge. Each of the two random graph distributions that we introduce below associates a
probability with every graph in this set.
G(n,p)
In the first case, the edges of a graph are regarded as a set of independent Bernoulli variables. If we
letXij denote the edge variable for the pair of nodes i andj, andp be the (uniform) probability that
the edge between i andj is present, then we can write Pr(Xij = 1) =p and hence Pr(Xij = 0) = 1-p.
Since the edge variables are independent, it is then easy to write down the probability of any
particular graphHof ordern and size m:
Pr (G =H) =pm(1 -p)
m*-m
where m* = n(n-1)/2 is the maximum number of edges in a graph of ordern. The set of all possiblegraphs of ordern and their corresponding probabilities is the random graph distributionG(n,p).
In the special case wherep = 0.5, the probability of each graphHon n nodes is:
Pr (G =H) = (0.5)m(0.5)m*-m = (0.5)m*
and hence every graph on n nodes is equiprobable. This distribution is often termed the uniform
random graph distribution, and denoted by U.
G(n,m)
The second random graph distribution associates nonzero probabilities only with graphs of ordern
and size m; further, every such graph is assumed to be equiprobable. Since there are n!/(m!(n-m)!)
distinct graphs in the class, the probability of any particular graph of ordern and size m is:
Pr (G =H) = m!(n-m)!/n!
The distribution G(n,m) may be regarded as the uniform random graph distribution on n nodes,
conditional on the property of having m edges. It may also be designated Ux++=m. More
generally, we can define a conditional uniform random graph distribution in terms of any graph
property Q(Bollobs, 1985). IfQis a subset of all possible graphs on n nodes, then the distribution
UQassigns equal probabilities (namely, 1/|Q|) to all graphs in the subset Qand zero probability to
all graphs not in Q. We term UQthe uniform random graph distribution conditional onQ.
Some results for simple random graphs
The field of random graphs in G(n,p) and G(n,m) has grown rapidly since its inception. Much work
has explored the features of various random graph classes, particularly as n becomes large, and the
way in which these features change, often very rapidly, as a function ofp. Bollobs (1998) contains
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an excellent introduction to the field, as does the review by Albert and Barabsi (2002); here we
illustrate just two aspects of this literature by considering the expected values of some statistics in
G(n,p) and by reviewing properties of a graph as a function ofp in G(n,p).
We begin by determining the expected number of cliques in a graph. Let Ys = Ys(G) be the number
of cliques of orders in the graph G. Then the expected value ofYs can readily be computed as:
E(Ys) = (n! /[s!(n-s)!]) pu
where u = s(s-1)/2 is the number of edges in a clique of orders (for example, see Bollobs, 1998).
In G(17,.25), for example, the expected number of cliques of order 3 is 10.6, and the expected
number of cliques of order 4 is 0.58. The graph of Figure 1 has 17 nodes and a density of 0.25 and
it is therefore interesting to compare the expected values forG(17,.25) with those observed for the
graph of Figure 1, namely, 34 cliques of order 3 and 18 cliques of order 4.
Indeed, ifFis any subgraph with s nodes and tedges, and YF is the number of subgraphs ofG that
are isomorphic to F, then the expected value ofYF can readily be shown to be:
E(YF) = (n! /[(n-s)!a]) pt
where a is the number of distinct ways in which the nodes of the graph Fcan be labeled with the
integers 1,2, , s to yield the same graph. For example, in the case of a cycle of orders, Fhas s
edges and s nodes and there are 2s distinct ways in which nodes can be labeled to yield the samegraph; hence:
E(YF) = (n! /[(n-s)!2s]) ps
The expected number ofinducedsubgraphs isomorphic to Fmay be similarly derived:
E(YF) = (n! /[(n-s)!a]) pt(1-p)s(s-1)/2 - t.
The latter formula may be used, for example, to compute the expected triad census for a graph of
given order and density.
The expression for the expected number of subgraphs isomorphic to Fin G(n,p) allows us to
explore features of random graphs as n tends to infinity. Since
E(YF) = (n! /[(n-s)!a]) ptnspt/a
it is clear that ifp = cn-s/t then E(YF) ct/a and the expected number of subgraphs isomorphic to F,
denoted by = ct/a, is a finite number. If, however,pns/t tends, respectively, to 0 or as n tends to
, then the probability that a random graph in G(n,p) contains at least one subgraph Fconverges to
0 or 1, respectively (e.g., Albert & Barabsi, 2002). Thusp = cn
-s/t
is a critical probability, belowwhich graphs with large n rarely contain F, above which they almost certainly do. An important set
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of results in random graph theory document the many graph properties that show this form of rapid
transition from being very unlikely to very likely as a function of the edge probabilityp.
Application of this approach allows us to infer for large n some expected features of random graphs
in G(n,p) as a function ofp, relative to n. Thus, for example, ifp < 1/n then almost every graph in
G(n,p) comprises a number of components, each without any cycles; ifp lies between 1/n and (ln
n)/n, then almost every graph has a so-called giant component(that is a component including a
large proportion of the nodes inN); and ifp > (ln n)/n, then almost every graph is connected (e.g.,
see Albert & Barabsi, 2002).
Random directed graph distributions may be similarly defined, though interest in them has largely
come from social scientists with applications to network data in mind. It is to this literature that wenow turn.
Applications of random graph and directed graph distributions to social network data
Before describing the application of random graphs to social networks, it is important to say
something about typical sources of social network data (eg Wasserman & Faust, 1993). A common
method of measuring social networks is to survey all members of a circumscribed population about
their ties. In this case, ties are typically directional and there may or may not be a limit imposed on
the maximum number of ties reported by each respondent. Occasionally in this case, it is fruitful to
consider the graph constructed from mutual ties only. For example, the graph of Figure 1 is the set
of mutual ties observed among the girls in a Grade 8/9 high school class, obtained in response to the
question: Who are your best friends in the class? Networks are also commonly inferred from
archival data, such as communication logs or membership or attendance lists. Less common
strategies include direct observation, and more elaborate survey techniques.
Application of directed random graph distributions
In the 1930s, the psychiatrist Jacob Moreno and colleagues reported using random directed graph
distributions to compute such quantities as the expected number of mutual ties (that is, i,jXijXji/2),
whereXij denotes the random variable for the tie from node i to nodej. Moreno and colleagues also
calculated properties of the indegree distribution in a random directed graph distribution (see, for
example, the very interesting historical account in Freeman, 2004).
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Consider, for example, the random graph distribution DG(n,p) with a fixed set ofn nodes and
uniform but unknown arc probabilityp; this is the directed graph analogue ofG(n,p). The expected
number of mutual ties for directed graphs in this distribution is n(n-1)p2/2, and the expected number
of nodes with indegree kis npk(1-p)n-1-k. If a social network with n nodes andx++ = i,jxij arcs has
been observed, then the arc probabilityp can be estimated from the network data asp* =x++/[n(n-
1)]. This estimate of the arc probability can be utilized to compute the expected number of mutual
ties and the expected number of nodes with each possible indegree k, on the assumption that the
observed network was generated from DG(n,p*). These expected values can be compared with the
observed number of mutual ties and the observed indegree distribution in the networkx. If the
observed values are markedly different from the expected ones, then, it can be argued, there is
reason to question the suitability of the assumption of independent random arcs with uniform
probability that underpinned the computation of expected values.
The computations by Moreno and colleagues revealed what would later become a very common
finding: that the observed number of mutual ties in an observed human social network is much
greater than the number expected on the basis of arc probability alone, and the indegree distribution
is more heterogeneous than expected, that is, there are more nodes with very low and very high
indegree than expected. These and similar findings suggest that tendencies towards mutuality and
heterogeneity in partner attractiveness are systematic features of observed social networkscomprising ties of affiliation.
Most importantly for our purposes here, Moreno introduced the idea of using random directed graph
distributions as null distributions. The features of this null distribution could be compared with
features of an observed network, a comparison enabling researchers to identify ways in which the
observed network appeared to be systematically different. In this early example and in many to
follow, it was important that the expected features of the null distribution could be derived
mathematically. Much later, when fast computers and more versatile simulation algorithms were
introduced, this restriction could be relaxed, but it was an important reason for the focus of early
applications on this null distribution approach. Moreover, although this early application
assumed very simple null distributions (such as, independent arcs with uniform tie probability),
more complex distributions were soon developed. Indeed, the strategy continues to be utilized in
new ways (e.g., Bearman et al, 2004; Pattison, Wasserman, Robins & Kanfer, 2000) and to be
rediscovered in new fields (e.g., Milo, Shen-Orr, Itzkovitz, Kashtan, Chlovskii & Alon, 2002). It
remains an important means by which some of the systematic structural properties of human social
networks can be and have been uncovered.
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Holland, Leinhardt and colleagues were responsible for developing a number of important
elaborations of this basic strategy. For example, Holland and Leinhardt (1975) computed the
expected mean vector and variance-covariance matrix for the triad census in the uniform random
directed graph distribution Umut,asym,null2
conditional on fixed numbers mut, asym and null of
mutual, asymmetric and null ties, respectively (mut= i,jxijxji/2, asym = i,j[xij(1-xji)+ xji(1-xij)], null
= i,j(1-xij)(1-xji)). In other words, they computed the expected distribution of the triad census
while conditioning on the dyad census. This allowed them to construct a test statistic for any linear
combination of triad counts, and hence assess whether the observed combination of triad counts is
in the upper or lower tail of the expected distribution. For example, they could test for the presence
oftransitivity (that is, the property that arcs from nodes i toj and fromj to kare accompanied by an
arc from i to k).
Similar calculations can be made for other distributions of possible interest, including the uniform
distribution U{xi+},mutconditional on the outdegrees of each node in the directed graph as well as
the number of mutual ties. Of course, for some desirable combinations, such as U{xi+},{x+i},mut,
the calculations are very difficult and have prompted alternative parametric approaches. In some of
these difficult cases, clever simulation strategies have been devised to circumvent the difficult
mathematics. For example, Snijders (1991) used an importance sampling approach to simulate
U{xi+},{x+i} and McDonald, Smith and Forster (in press) have described a Markov Chain Monte
Carlo algorithm to simulate the distribution U{xi+},{x+i}, mut.
Biased nets
One other early probabilistic approach deserves mention. In a series of papers, Rapoport and
colleagues developed the theory ofbiased nets, that is, random networks with biases towards
symmetry, transitivity and other features characteristic of observed social networks (Rapoport,
1957). Although a full and satisfactory mathematical treatment proved elusive, Rapoport and
colleagues employed their conceptualization of biased nets to conduct some illuminating studies of
the connectivity structure of a large friendship network (e.g., Rapoport & Horvath, 1961). In part,
their work can be seen as the intellectual precursor to the more general probabilistic developments
described below, but a different framing of the biases has proved more useful.
2In fact, Holland and Leinhardt (1975) termed this the U|MAN distribution, but we have attempted to keep notation
consistent in the chapter.
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The p1 model
Comparison of observed social networks with random directed graph distributions consistently
revealed a greater than expected number of mutual ties and greater than expected degree
heterogeneity. As a consequence, it was felt desirable to compare observed networks with graph
distributions that resembled observed networks in these fundamental respects. The problem of
satisfactorily simulating the random graph distribution conditional on the number of mutual ties and
the in- and out-degree sequences is arguably still not resolved. In the meantime, Holland and
Leinhardt (1981) developed an alternative approach: a probability model that parametrised these
tendencies. This model was an important step towards the development of a more general
framework.
Thep1 model developed by Holland and Leinhardt (1981) assumes independent dyadsDij = (Xij,Xji).
The distribution of the entire networkX = [Xij] can then be determined by specifying the probability
of each possible dyadic form forDij since the probability of the entire networkX is the product of
the dyad probabilities. The individual dyad probabilities can be expressed in terms of the
probability of occurrence of a mutual dyad, an asymmetric dyad, and a null dyad. Thus, we define:
Pr(Dij=(1,1)) = mij= mji, Pr(Dij=(1,0)) = aij, and Pr(Dij=(0,0)) = nij = nji
where mij + aij + aji + nij = 1 for all ij.
The resulting probability distribution
Pr (X = x) = mijXijXji aijXij (1 Xji) nij(1-Xij) (1 Xji)i
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ij = + i + j forij.
The resulting model is termed thep1 model:
p1(x) =Pr(X = x) = Kexp[i,jXijXji + X++ + iiXi+ + iiX+i]
The parameters and can be interpreted as uniform reciprocity and density parameters, and the
node-dependent parameters i and i reflect the expansiveness and attractivesness, respectively, of
each node i.
The development of thep1 model was an important step in probabilistic network theory, not least
because much of the machinery of statistical modeling could be brought to bear on the problem of
assessing model adequacy. The model could be estimated from data, and its goodness of fit could
be subjected to careful scrutiny, as Breiger (1981) demonstrated. Such scrutiny led to the
recognition that observed networks often exhibited structural properties not captured by the
parameters of thep1 model, and spawned the development of two important lines of further model
development.
Latent variable models
The first line of development is a series of latent variable models, in which the assumption of
independent dyads is replaced by an assumption of independent dyads conditional on unobserved
variables representing some potential underlying structure.
One such model was inspired by the concept of structural equivalence in a graph (Lorrain & White,
1971). Two nodes are structurally equivalent if they have identical patterns of relationships to other
nodes. The concept of structural equivalence has been very influential in the social networks
literature because it can be used to represent the idea that two actors have the same social position
in a network, that is, that they are indistinguishable from a relational point of view.
Formally, two nodes i andj are structurally equivalentin a directed graph G with adjacency matrix
x ifxik=xjkandxki =xkj for all nodes ki,j inN. Structurally equivalent nodes can be partitioned
into blocks. Nowicki and Snijders (2001) assumed that the blocks to which nodes belong are
unobserved. They defined a set of independent and identically distributed latent random variables
Z = [Zi] whereZi denotes the blockof node i, and Pr(Zi = k) = k. They assumed the dyadsDij =
(Xij,Xji) to be conditionally independent given the blocks and the probability that a dyad has a
particular relational form to depend only on the (unobserved) blocks of the nodes. In other words:
Pr(Dij = a|Z = z) = a(zi,zj)
where:
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a is a vector of possible values for the dyad, with a {(1,1), (1,0), (0,1), (0,0)} for a
directed graph and a {(1,1), (0,0)} for a graph; and
a(zi,zj) is the block-dependent probability of observing the vectora.
In this model, two nodes i andj are stochastically equivalentif they belong to the same block and
hence the same dyad probabilities (Pr(Dik= a|Z = z) = Pr(Djk= a|Z = z)) for all nodes k. Since the
dyads are assumed to be conditionally independent given the blocks Z, the joint distribution of the
Dij given Z is the product of the conditional dyad probabilities. Nowicki and Snijders (2001)
developed a Bayesian approach to estimation of and , and hence computation of the posterior
probabilities that any pair of nodes are in the same block and that a dyad has any particular
relational form.
Several other important latent variable models have been developed for particular types of social
networks that are likely to reflect some form of proximity among actors, such as friendship or
collaboration. In cases such as these, it may be reasonable to assume that tie probabilities are
monotonically related to proximity in a latent space. For example, Hoff, Raftery and Handcock
(2002) proposed a model which assumes that nodes have latent locations in some low-dimensional
Euclidean space and that, given these latent locations, tie variables are conditionally independent.
Schweinberger and Snijders (2003) developed a similar model based on an ultrametric rather thanEuclidean space; in their model, every pair of nodes is associated with an unobserved distance in an
ultrametric space corresponding to a discrete hierarchy of settings, and tie probabilities are
conditionally independent, given these latent ultrametric distances. (Distances in an ultrametric
space satisfy the ultrametric inequality, that is d(i,j) max{d(i,k), d(j,k)} for any triple of nodes i,j,
k.) They developed approaches for estimating the unobserved ultrametric distances. Handcock,
Raftery and Tantrum (2005) have recently extended Hoff et als (2002) model by assuming that the
latent locations are drawn from a finite mixture of multivariate normal distributions, each of whichrepresents a different group of nodes.
Markov random graphs
The second recent line of development has been to build probabilistic network models in which
conditional dependencies among tie variables are permitted. This work began with the recognition
by Frank and Strauss (1986) that a general approach for modeling interactive systems of variables
(Besag, 1974) could be usefully applied to the problem of modeling systems of interdependent
network tie variables on a fixed set of nodes. This was an important step because it permitted
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models to go beyond the limiting assumption of dyad-independence in a quite general way. Frank
and Strauss (1986) introduced aMarkov dependence assumption for network tie variables: two
network tie variables were assumed to be conditionally independent, given the values of all other
network tie variables, unless they had a node in common. Thus, whereas a tie between nodes i andj
was assumed to be conditionally independent of ties involving all other distinct pairs of nodes kand
l, it could be conditionally dependent on any other ties involving i and/orj.
Assumptions about which pairs of tie variables are conditionally dependent, given the values of all
other tie variables, can be represented as a dependence graph. The node set of the dependence
graphD is the set of tie variables {Xij} and two tie variables are joined by an edge inD if they are
assumed to be conditionally dependent, given the values of all other tie variables. In the case of
G(n,p),D is an empty graph since all pairs of variables are assumed to be mutually independent. Inthe Markov case, the variableXij is connected toXikandXjkfor all ki orj and the dependence
graph is connected. Figure 5 shows the Markov dependence graph for a random graph of order 4.
--------------------------------
Insert Figure 5 about here
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As Frank and Strauss originally outlined, the consequences of any proposed assumptions about
potential conditional dependencies among network tie variables can be inferred from the
Hammersley-Clifford theorem (Besag, 1974). The theorem establishes a model for the interacting
system of tie variables in terms of parameters that pertain to the presence or absence of certain
configural forms in the network. The model, known as an exponential random graph model, takes
the general form:
Pr(X = x) = exp(AAzA(x))/
where:
A is a subset of tie variables (defining a potential networkconfiguration);
A is a model parameter associated with the configurationA (to be estimated) and is
nonzero only if the subsetA is a clique in the dependence graphD;
zA(x) = XijAxij is the sufficient statistic corresponding to the parameterA, and
indicates whether or not all tie variables in the configurationA have values of 1 in the
networkx; and
is a normalizing quantity.
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In order to reduce the number of model parameters, Frank and Strauss (1986) introduced a
homogeneity constraint that parameters for isomorphic configurations are equal. With this
constraint, there is a single parameter [A] for each class [A] of isomorphic configurations that
correspond to cliques in the dependence graph. The sufficient statistic in the model corresponding
to the class [A] is then
z[A](x) = A[A]XijAxij,
that is, a count of all observed configurations in the graph x that are isomorphic to the configuration
corresponding toA. For example, in the case of a homogeneous Markov random graph, it is readily
seen that cliquesA in the dependence graphD correspond to graph configurations that are edges,
stars and triangles (see Figure 6) and the model therefore takes the form:
Pr(X=x) = exp(L(x) + kkSk(x) + T(x))/
whereL(x), Sk(x) and T(x) are the number of edges, k-stars (2 kn-1) and triangles in the
networkx, and , k(2 kn-1), and are corresponding parameters.
In many circumstances, the parameters may be interpreted by observing that if a configuration class
[A]has a large positive (or negative) parameter in the model, then the presence of many
configurations in the class enhances (or reduces) the likelihood of the overall network, net of the
effect of all other configurations. It should be noted, though, that the function relating the value of
one of the models parameters, say [A] to the expected value of the corresponding sufficient
statisticz[A](x) may be markedly nonlinear and exhibit a sharp and rapid transition from lower
average counts to higher average counts with a relatively small change in the parameter (holding
constant the values of all other model parameters). For example, the expected number of triangles
in a graph as a function of the triangle parameter is shown for a graph on 17 nodes in Figure 7.
The values of the parameters , 2 and 3 are fixed at -1.2558, -0.0451, and -0.1084, respectively,
and takes values in the range [0.05,1.70]. Figure 7 shows the distribution of the triangle statistic
in the form of a box plot for each value of. It can be seen that for low values of small increases
in are associated with small and steady increases in the triangle statistic. As approaches 1.40,
though, the impact of small changes in increases rapidly in magnitude, and there is a sharp
transition to a higher value of the triangle statistic. Near the point of transition, the triangle statistic
may take values typical of the graphs on either side of this apparent threshold. This form of
nonlinear relationship is common, and the location of this threshold and the sharpness of the rise in
the region of greatest sensitivity are likely to depend on other parameter values.
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-----------------------------------------
Insert Figures 6 and 7 about here
-----------------------------------------
It is important to emphasise that even though this model is well-understood in the case where only
the parameter is nonzero (since this is just the model G(n,p) withp = exp()/[1+exp()]), more
complex instantiations can be seen as models for self-organizing network processes (Robins,
Pattison & Woolcock, 2005). Robins et al have demonstrated that specific sets of parameter values
for the homogeneous Markov model can characterize very diverse network structures, including
small worlds, caveman worlds, long-path worlds, and so on.
For some parameter values, the model may accord very high probability to a small set of graphs andvery low probability to the rest, as Handcock (2004) and Snijders (2002) have demonstrated.
Handcock (2004) termed these models near-degenerate. For detailed investigation of the behaviour
of specific models, see Handcock as well as Park & Newman (2004) and Burda, Jurkiewicz and
Krzywicki (2004).
Model simulation
In order to understand properties such as near-degeneracy of the exponential random graph model
Pr(X = x) = exp(AAzA(x))/, it is helpful to be able to simulate it efficiently (that is, to draw
graphs x with probability Pr(X = x)), and this generally means circumventing the need to compute
the normalizing quantity , since is a function ofall graphs in the distribution. As Strauss (1986)
and others have observed, theMetropolis algorithm can be used for this purpose. The algorithm
sets up a Markov chain on the space of all possible graphs of ordern in such a way that the Markov
chain has the model as its stationary distribution. It may be described as follows:
1. begin with some graph x;
2. at each step, select an edge at random, say the (i,j) edge, and let x be the graph that is
identical to x except that the edge from i toj is switched to absent if it is present in x, or to
present if it is absent in x;
3. replace x by x with probability min[1,exp{AA(zA(x) -zA(x)}];
4. return to step 2, unless some specified target number of steps has been taken.
To sample from Pr(X = x), it is usual to begin sampling graphs from the chain after some initial
number of steps has been completed (the burn-in period); graphs are then sampled at a rate that may
depend on n. Many variations of this approach may also be used; a valuable discussion may be
found in Snijders (2002).
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Estimation of model parameters
In many settings, primary interest lies in estimating exponential random graph model parameters
from observed network data. For example, given the graph of Figure 1, there may be interest in
estimating the parameters of a Markov model from which it might have been generated. In the
early applications of these models to observed data, an approximate form of estimation known as
pseudo-likelihood estimation was often used (Strauss & Ikeda, 1990; Wasserman & Pattison, 1996)
even though the properties of the estimates were not well understood. Initial attempts to apply the
very promising approach of Markov Chain Monte Carlo maximum likelihood estimation
(MCMCMLE) were not always successful because the properties of models under consideration
were not always fully appreciated, as Snijders (2002) and Handcock (2004) demonstrated.
However, with a growing understanding of model properties, and more careful attention to modeladequacy, substantial progress has now been made in implementing MCMCMLE approaches (see
Snijders, 2002; Handcock et al, 2004).
As an example of MCMCMLE, we estimate the parameters of the model
Pr(X=x) = exp(L(x) + 2S2(x) +3S3(x) + T(x))/
for the mutual friendship network of Figure 1 using the approach proposed by Snijders (2002). The
resulting estimates and estimated standard errors for the parameters , 2, 3, and
shown in Table
23. Also displayed in Table 2 are convergence tstatistics, computed as the difference between the
observed value of the sufficient statistic for a parameter and its average simulated value, divided by
the standard deviation of simulated values. If the estimated value is indeed the MLE, the simulated
values should be centred on the observed value, and the tstatistics should all be small, preferably
below 0.1 (Snijders, 2002). It can be seen from Table 2 that the tstatistics satisfy this requirement.
Although the edge, 2-star and 3-star parameters are negative and within about one standard error of
zero, the triangle parameter is positive and approximately 7 times its estimated error. This suggests
that, other graph features (edges, 2-stars and 3-stars) being equal, graphs with more triangles are
more likely. That such a model is needed for the graph of Figure 1 is consistent with the earlier
computations based on G(17, 0.25).
--------------------------------
Insert Table 2 about here
--------------------------------
3 The estimation was conducted using PNet (Wang, Robins & Pattison, 2006), an implementation of the estimation
approach in Snijders (2002) available at: http://www.sna.unimelb.edu.au/pnet/download.html.
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Goodness of fit
A good statistical model should not be unnecessarily complex but it should be adequate: that is, the
data should resemble realizations from the model in many important respects. We can assess model
adequacy by comparing the observed network to graphs generated by the model in features that are
not necessarily parameterised within the model. What is important in such comparisons is very
much a function of the modeling context, but there are often good reasons to require that the model
captures the degree of clustering in a network, the distribution of degrees, and the connectivity
structure that is represented by the geodesic distribution (e.g., Goodreau, in press; Robins, Snijders,
Wang, Handcock & Pattson, in press). It is important to note that only some of these characteristics
need be associated with model parameters; others might be seen as consequences of these
parameterized tendencies.
For example, if we simulate the model
Pr(X=x) = exp(L(x) + 2S2(x) +3S3(x) + T(x))/
using the parameter estimates in Table 2, we can not only compare the observed graph to the
simulated graph in terms of its sufficient statistics (namely, the number of edges, 2-stars, 3-stars and
triangles), but we can also make the comparison in relation to any unmodelled network
characteristic, such as the number of nodes of degree 4 or more, the number of geodesic distances
of length 3, and so on.
Table 3 summarises these comparisons for the graph of Figure 1 and the parameter estimates of
Table 2. It can be seen that the tstatistics are all less than 1 for:
the local clustering coefficient(the average across all nodes i of the proportion of pairs
of nodesj and kincident with i (xij = 1 =xik) that are themselves connected (xjk=1);
the global clustering coefficient(the proportion of the triples of nodes {i,j,k} withxij = 1
=xikfor whichxjk=1);
the standard deviation of the degree distribution; and
the skewness coefficient of the degree distribution.
The observed graph, in other words, exhibits levels of clustering and degree heterogeneity that fall
within the envelope of values expected for the model. The first, second and third quartiles of the
observed geodesic distribution are 1, 3 and 4, respectively; the median values for the distribution of
these quartiles across simulations were 2, 2 and 4, suggesting that the model is associated with
somewhat more homogeneous inter-node distances than the data. In Figure 8, the distributions of
the number of edges, 2-stars, 3-stars and triangles for the Markov random graph model with these
parameter values are shown. While the mean of each distribution is close to the observed value for
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the Figure 1 graph, as expected, it can be seen that the distributions are positively skewed. Indeed,
Figure 7 shows the impact on one of these statistics, the number of triangles, of changing its
corresponding parameter value while holding all other parameters constant. It can be seen from
Figure 7 that the estimated value of 1.4438 is very close to the point of transition between low and
high values of the triangle statistic, and the positively skewed distribution of the triangle statistic is
consistent with the estimated value of being just below this point.
-----------------------------------------------
Insert Figure 8 and Table 3 about here
-----------------------------------------------
Related model parameters
In the Markov model just fitted, parameters for 2-stars and 3-stars were included, but parameters forhigher order stars (4-stars, 5-stars and so on) were assumed to be zero. Arguably, fitting higher
order star parameters might be desirable, because more star parameters will lead to better
characterizations of the degree distribution for the network. Indeed, Snijders, Pattison, Robins and
Handcock (2006) proposed that all star parameters be used, but also imposed an hypothesis about
the relationships among stars parameters. Specifically, they assumed
k+1 = (-1/)kfork>1 and 1 a (fixed) constant,
an hypothesis they termed the alternating k-star hypothesis. It follows from this hypothesis that:kkSk(x) = [k(-1)
kSk(x)/
k-2]2
and hence that the entire set of star-like terms in the model can be captured by a single star
parameter (2) with a single alternating k-starstatistic:
S[](x) = k(-1)
kSk(x)/
k-2
It is of course an empirical matter whether this is an appropriate hypothesis to make. This
expression can be simplified to yield a simpler form of the alternating k-star statistic:
S
[]
(x) =
2
i{(1-1/)k(i)
+ k(i)/ -1},recalling that k(i) denotes the degree of node i.
4
Fitting the model
Pr(X=x) = exp(L(x) + 2 S[](x))/
to the friendship network in Figure 1 yields estimates (standard errors) of -2.7454 (1.3794) and
0.4822 (0.4098) for and 2, respectively (as before, is an edge parameter andL(x), the number
of edges in the networkx, is its sufficient statistic). Although this model does a reasonable job in
4 Hunter and Handcock (in press) proposed an alternative statistic based on geometrically weighted degree statistics; the
resulting model is equivalent provided that the edge parameter is included.
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reproducing the standard deviation and the skewness coefficient for the degree distribution, not
surprisingly, it does a poor job in recovering network clustering. We present a better fitting model
below.
In some applications to date (eg Goodreau, in press; Snijders et al, 2006), a fixed value of (such as
2) has been assumed; Hunter and Handcock (2006) have shown that can be treated as a variable
within a curved exponential family model, and have developed an associated estimation method.
Realisation-dependent models
A critique of the Markov dependence assumption led Pattison and Robins (2002) to construct a
more general class of realisation-dependent network models. They argued that conditionaldependencies among tie variables may emerge from the network processes themselves, with new
dependencies created as network ties are generated. For instance,Xij andXkl might become
conditionally dependent ifthere is an observed tie between, say,j and k. Baddeley and Mller
(1989) termed such models realisation-dependent.
The 4-cycle hypothesis
Snijders et al (2006) argued that, in addition to the Markov assumption, two network tiesXij
and
Xkl might be conditionally dependent in the case that there is an observed tie between, say,j and
kandbetween l and i, that is, if the presence of a tie from i to j and from kto l would create a 4-
cycle in the graph. The rationale for this assumption is that a 4-cycle is a closed structure that
can sustain mutual social monitoring and influence, as well as levels of trustworthiness within
which obligations and expectations might proliferate (e.g., Coleman, 1988).
Snijders et al showed that this assumption led to additional nonzero parameters in an exponential
random graph model, including those referring to collections of 2-paths with common starting
and ending nodes, and collections of triangles with a common base (see Figure 9). We define a
k-2-path to be a subgraph comprising two nodes, i andj, and a set ofkpaths of length 2 from i to
j through distinct intermediate nodes m1, m2, , mk. A k-triangle is a subgraph comprising two
connectednodes, i andj, and a set ofkpaths of length 2 from i toj through distinct intermediate
nodes m1, m2, , mk. If we let kbe the model parameter associated with a k-2-path and kthe
parameter associated with a k-triangle, we can entertain assumptions about the relationships
among related parameters (as in the case ofk-stars earlier), namely:
k+1 = -k/, and
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k+1 = -k/.
As for the star parameters, this is just an hypothesis, and its adequacy needs to be assessed.
Under this assumption, the statistics:
U[](x) = k(-1)
kUk(x)/
k-2
and
T[](x) = k(-1)
kTk(x)/
k-2
become single statistics associated with the parameters 1 and 1, respectively, where Uk(x) and
Tk(x) are the number ofk-2-paths and k-triangles in the networkx. It should be noted that the
value of need not be the same for each statistic; as before, Hunter and Handcock have shown
how to estimate these parameters.
The parameter estimates presented in Table 4 are for a model fitted to the mutual friendship
network of Figure 1. The positive 1 estimate suggests that networks with relatively many
triangles are more likely, other statistics being equal, with the cumulative impact of multiple
triangles with a common base pair of nodes diminishing as the number of such triangles
increases. Likewise, the negative 1 estimate suggests that networks with relatively few 2-paths
among a pair of non-connected nodes are more likely, other statistics being equal. Both of these
effects are consistent with a pressure towards closure for mutual friendship ties.
The goodness of fit for this model is summarised in Table 5. The median values of the quartiles
of the geodesic distribution for the random graph distribution simulated from Table 4 parameter
estimates are 2,3 and 5, respectively, suggesting better recovery of short distances than the
Markov model, though not of longer ones. Overall, the model of Table 4 appears to do a
reasonably good job of characterising the features of the mutual friendship network.
-------------------------------------------------------
Insert Figure 9 and Tables 4 and 5 about here-------------------------------------------------------
Directed graph models
The derivation of similar classes of models for directed graphs is, in principle, very similar to the
derivation of models for their nondirected counterparts. Directed graphs give rise, however, to
substantially more complicated parameterizations, as a comparison between triadic forms in graphs
and directed graphs quickly suggests. There are, as a result, some subtleties to the development of
models in the directed graph case (see Robins, Pattison and Wang, 2006, for further details).
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Exogenous covariates
The general modeling framework can readily accommodate covariates at the node or dyad level
and, of course, if such covariates are regarded as important influences on network tie formation,
then they should be included in models for the network. For example, a general and systematic
approach to the inclusion of node-level covariates has been outlined by Robins, Elliott and
Pattison (2001), who extended the dependence graph formulation described earlier to include
directed dependence relationships from exogenous node-level variables to endogenous tie
variables. These developments offer an important means for exploring a wide range of
interesting interactions among actor-level and network tie variables, including homophily effects
(e.g., McPherson, Smith-Lovin & Cook, 2001) and the effects of spatial locations (e.g., Butts,
2003; Wong, Pattison & Robins, 2006).
Extensions
There are a variety of ways in which the models just described may be extended to incorporate
richer data forms, including multiple networks, longitudinal data, and changing node and tie sets.
In addition, some initial progress has been made on the problem of dealing with missing data. We
do not have space for a full account of these interesting and important developments, but point to
some key developments in each case.
The development of probabilistic models for graphs and directed graphs has been loosely shadowed
by the construction of a parallel, albeit generally later, set of models for multiple networks
measured on a common node set. Multivariate exponential random graph models are described by
Pattison and Wasserman (1999; see also Koehly & Pattison, 2005).
Although networks are often measured at a single point in time they are, in reality, dynamic entities
and there is considerable interest in the processes that underpin their evolution. An important line
of work has developed continuous-time Markov process models for network evolution (eg see
Snijders, 2001). More recently, this framework has been extended to accommodate the possibility
of co-evolutionary mechanisms by which network tie change depends on and contributes to change
in node attributes (Snijders, Steglich & Schweinberger, in press).
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Concluding comments
The field of probabilistic network theory has progressed rapidly in the last ten years and, as
Goodreau (in press) and Robins, Snijders et al (in press) have cogently demonstrated, it is now
possible to build plausible models for many small and large social networks. Undoubtedly,
experience with the current generation of realisation-dependent network models will lead to further
improvements in model specification, and a clearer understanding of how the content of network
ties and the contexts in which they are observed might inform model building. Perhaps most
importantly, though, the field has now advanced to the point where the promise of a step-change in
our understanding of social processes on networks and their consequences might be realised.
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Table 1. Adjacency matrix for graph of Figure 1
-------------------------------------------
1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7
-------------------------------------------
1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 12 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0
3 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0
4 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
6 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
7 0 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0
8 0 1 1 1 0 0 1 0 1 1 1 0 0 0 0 0 0
9 0 1 0 1 0 0 1 1 0 1 1 0 0 0 0 0 0
10 0 1 1 0 0 0 1 1 1 0 1 0 0 1 0 0 0
11 0 1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0
12 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 113 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
14 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0
15 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1
17 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0
-------------------------------------------
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Table 2. MCMCMLEs for Markov model of the graph of Figure 1
-------------------------------------------------
Parameter estimate s.e. t
-------------------------------------------------
edge -1.2558 1.3561 0.047
2-star -0.0451 0.3551 0.060
3-star -0.1084 0.0974 0.073triangle 1.4438 0.2073 0.058
-------------------------------------------------
Table 3. Goodness of fit for Markov model for the mutual friendship network (Figure 1)-------------------------------------------------------------------------------
Simulated
Observed Mean std dev t
-------------------------------------------------------------------------------
edges 34 34.07 8.91 -0.0077
2-stars 139 138.35 77.45 0.0084
3-stars 181 178.05 156.49 0.0189
triangles 30 29.34 28.39 0.0231
std dev degrees 2.09 1.71 0.46 0.8296
skew degrees 0.08 -0.26 0.62 0.5480
global clustering: 0.65 0.54 0.21 0.5246
mean local clustering 0.64 0.47 0.19 0.4917
variance local clustering 0.14 0.10 0.04 0.9899
--------------------------------------------------------------------------------
Table 4. MCMCMLEs for realization-dependent exponential random graph model for the mutual
friendship network (Figure 1)
------------------------------------------------
Parameter estimate s.e. t
------------------------------------------------
edge -0.0354 1.7851 -0.037
2-star -0.0520 0.1094 -0.038
k-star 0.0674 0.8689 -0.040
k-triangles 0.7250 0.3159 -0.043
k-2-paths -0.5583 0.1727 -0.025
------------------------------------------------
Table 5. Goodness of fit of realisation-dependent model for the mutual friendship network
Simulated
Statistic Observed Mean std dev t
-------------------------------------------------------------------------
Edges 34 34.44 9.33 -0.047
2-stars 139 145.13 94.54 -0.065
k-stars 77.8 79.73 34.68 -0.054
k-triangles 46.0 47.01 28.39 0.023
k-2-paths 83.3 85.30 30.68 -0.064
Std Dev degrees 2.09 1.77 0.60 0.522
Skew degrees 0.08 -0.17 0.46 -0.548
Global Clustering: 0.65 0.55 0.14 0.696
Mean Local Clustering 0.64 0.50 0.15 0.439
Variance Local Clustering 0.14 0.09 0.04 1.207-------------------------------------------------------------------------
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Figure 1. A graph on 17 nodes (mutual friendship network)
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Figure 2. A directed graph on 37 nodes (reported collaboration network)
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8.006.004.002.000.00
degree
4
3
2
1
0
Frequency
Mean =4.00
Std. Dev. =2.15058
N =17
Degree distribution for graph of Figure 1
Figure 3. The degree distribution for the mutual friendship graph
Figure 4. Geodesic distribution for the mutual friendship graph
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Figure 5. Dependence graph for a Markov random graph on 4 nodes
edge k-star triangle
k
nodes
Figure 6. Markov model configurations: edges, stars and triangles
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Figure 8. Distribution of edge, 2-star, 3-star and triangle statistics in the Markov random graph
distribution with parameters: = -1.2558, 2 = -0.0451, 3 = -0.1084, = 1.4438.
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knodes
knodes
knodes
Figure 9. The k-star, k-2-path and k-triangle
top related